Question

if b is the midpoint of $overline{ac}$, and $ac = 8x - 20$, find $bc$.
$3x - 1$
$b$
$c$

Explanation:

Step1: Use mid - point property

Since $B$ is the mid - point of $AC$, then $AB = BC$ and $AC=AB + BC = 2BC$. Given $AB = 3x - 1$ and $AC = 8x - 20$, we have $2(3x - 1)=8x - 20$.

Step2: Solve the equation for $x$

Expand the left - hand side: $6x-2 = 8x - 20$.
Subtract $6x$ from both sides: $-2=8x - 6x-20$.
Simplify: $-2 = 2x-20$.
Add 20 to both sides: $-2 + 20=2x$, so $18 = 2x$.
Divide both sides by 2: $x = 9$.

Step3: Find the length of $BC$

Since $BC=AB = 3x - 1$, substitute $x = 9$ into the expression.
$BC=3\times9 - 1=27 - 1=26$.

Answer:

$26$